1. Linear Functions

2. Definition
A function f is linear if its domain is a set of numbers and it can be expressed in the form
where m and b are constants and x denotes an arbitrary element of the domain of f.

3. Change and Rate of Change

4. Definition
If x1 and x2 are distinct members of the domain of f, the change in f from x1 to x2 is f(x2) – f(x1). The rate of change of f over the interval from x1 to x2 is

5. Notation
Let Dx = x2 – x1 denote the change in x. Let Df =f(x2) – f(x1) denote the change in f.
The rate of change is the ratio

6. Exercise
For real numbers x, let Find the change in f from x1 = 1 to x2 = 4.
Find the rate of change of f over the interval from 0 to 3 .
Find a general formula for the rate of change over the interval from x1 to x2 for any x1 and x2.

7. A Characterization of Linear Functions
A function from the real numbers to the real numbers is linear if and only if its rate of change is the same for all intervals. If so, the rate of change is the constant m in the formula

8. Graphs of Linear Functions
Straight Lines

9. Two distinct points
in the plane determine one and only one straight line

10. Point-Slope Form
Let be two distinct points in the plane.
Case 1:
Set
(slope)
Equation: or

11. Case 2:
Equation: x = c.

12. Point-Slope Form
Suppose it is known that a line passes through the point with coordinates and that it has slope m. Then the equation of the line is

13. Slope Intercept Form y = f(x) = mx + b
m = rate of change of f = slope of the line = tangent of angle between the x-axis and the line
b = f(0) = y-intercept of the line

14. Geometrical Interpretation

15. The Symmetric Form
Slope-intercept and point-slope forms cannot handle vertical lines in the xy plane.
Symmetric form does not select one variable as the independent variable and the other as the dependent variable. c, d, and e are constants.

16. Exercise The graph of a linear function is the line whose equation is
What is the rate of change of f? What are f(0) and f(-2)?

17. Systems of Linear Equations

18. General Form of a Linear System of Two Equations in Two Unknowns
Equations in Symmetric
Form of Two Straight Lines

19. Three Possibilities for Solutions
The lines are not parallel and intersect in one and only one point. That is, there is one and only one solution of the system.
The lines are distinct but parallel and do not intersect. There are no solutions.
The equations represent the same straight line. There are infinitely many solutions, one for each point on the line.

20. Examples: 1. 2. 3.

21. The Coefficient Matrix

22. The Determinant of the Coefficient Matrix
The number

23. Relationship of the Determinant to the Question of Solutions
The linear system has a unique solution if and only if the determinant is different from zero.

24. Cramer’s Rule
Not necessarily the best
method of solution.

25. Exercise
Solve
Answer: x=3/7, y=2/7

26. Inverses of Linear Functions

27. Example
Given y, solve for x:

28. Example (continued) The equation
defines x as a linear function of y. This function is called the inverse of the original function. We write

29. Equivalence The two equations and
are equivalent. One is satisfied by a pair (x,y) if and only if the other is.

30. General Expression for the Inverse Function
If f (x) = mx + b and m≠0, then
Note: The slope of the inverse function is the reciprocal of the slope of the original function.

31. The Graphs of the Function and Its Inverse